Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

ALIKAWA, Hidehisa

J. Math. Soc. Japan, Tome 59 (2007) no. 1, p. 151-177 / Harvested from Project Euclid

Résumé

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras $\mathfrak g$ to the fixed point subalgebras $\mathfrak g^{\sigma}$ of outer automorphisms $\sigma$ . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras $(\mathfrak g, \mathfrak g^{\sigma})$ includes an exceptional symmetric pair $(E_6, F_4)$ and also a non-symmetric pair $(D_4, G_2)$ as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of $\mathfrak g$ in terms of the subalgebras $\mathfrak g^{\sigma}$ on one hand, and the length function on the other hand.

Détail

Publié le : 2007-01-14
Classification:  branching rule,  multiplicity free,  outer automorphism,  graph automorphism,  semisimple Lie groups,  characters of Lie groups,  17B10,  05E15